Novel Nonconforming Finite Element Methods for Maxwell's Equations

这个项目的研究是基于最近发现的PI，即麦克斯韦方程的数值解可以基于使用函数空间的变分公式，其中实施了散度自由条件。这是通过将不可压缩流体流动的经典非协调有限元与间断Galerkin方法的技术相结合而实现的。在这种新方法中，时间调和(频域)Maxwell方程的边值问题被求解为椭圆型问题，而对源(确定性)问题和本征问题(腔体共振问题)的新的非协调有限元方法的性能与计算力学的经典有限元方法的性能相当。特别地，离散本征值既没有伪模，也没有非物理零本征值。所提出的研究将使用这种新的方法设计和分析许多新的麦克斯韦方程和麦克斯韦本征问题的格式。研究结果将为天线、雷达传感器、波导、光子晶体、磁阻传感器和粒子加速器等电磁器件的设计和分析提供强大的计算工具，应用于电信、集成光学、激光、高能物理、等离子体物理和无损损伤检测等不同领域。

The research in this project is based on the recent discovery of the PIs that numerical solutions of Maxwell's equations can be based on variational formulations that use function spaces where the divergence free condition is enforced. This is made possible by combining classical nonconforming finite elements for incompressible fluid flows and techniques from discontinuous Galerkin methods. In this new approach the boundary value problems of the time-harmonic (frequency-domain) Maxwell's equations are solved as elliptic problems, and the performance of the new nonconforming finite element methods for both the source (deterministic) problem and the eigenproblem (cavity resonance problem) is comparable to the performance of classical finite element methods for computational mechanics. In particular the discrete eigenvalues have neither spurious modes nor nonphysical zero eigenvalues. The proposed research will design and analyze many novel schemes for the Maxwell equations and the Maxwell eigenproblem using this new approach. Fast solvers (multigrid and domain decomposition methods) and adaptive algorithms will also be developed, with applications to related electromagnetic problems.The results of the proposed research will provide powerful computational tools for the design and analysis of electromagnetic devices such as antennas, radar sensors, waveguides, photonic crystals, magnetoresistive sensors and particle accelerators, with applications to diverse areas such as telecommunications, integrated optics, lasers, high energy physics, plasma physics, and nondestructive damage detection.